Appalachian set theory

September 9, 2006

Carnegie Mellon University

9:30 a.m. to 6 p.m. (with breaks) in Baker Hall A53 (Click for campus map)


Paul Larson : "An introduction to Pmax forcing"

The partial order Pmax, invented by W. Hugh Woodin, is one a family of forcings which produce models of ZFC when applied to models of determinacy. The partial order is given by a directed system of models of ZFC, and grew out of Woodin's proof that the existence of a measurable cardinal plus the saturation of the nonstationary ideal on ω1 implies the failure of the Continuum Hypothesis. The power set of ω1 in the Pmax extension is maximal in that it satisfies all Pi2 sentences whose forceability is implied by large cardinals. In this way the structure P(&omega1) in the Pmax extension is very similar to the one given by Martin's Maximum. Advantages of the Pmax approach include reduced large cardinal hypotheses (at the level of determinacy hypothesis) and avoidance of iterated forcing technology. Variations of Pmax are very useful for consistency results on P(ω1); some produce models in which the Boolean algebra P(ω1)/NSω1 is &omega1-dense. Larson and Todorcevic's consistency proof for the statement that every compact space with T5 square is metrizable grew out of another Pmax variation.

This one-day course will have two aims. The first is to start the participants along the shortest path towards applying Pmax in their own work. The second is to prepare them for reading Woodin's book on Pmax, The axiom of determinacy, forcing axioms and the nonstationary ideal, whose technical prerequisites are fairly high. We will start with the fundamental combinatorial arguments underlying Pmax, and at some point shift gears to a higher-level approach. We will black box the most technical issues in the basic analysis, the existence of Pmax conditions, though the proofs will be made available. Taking this step for granted, most of the Pmax analysis can be carried out using techniques which we should be able to present in full. The main reference will be my expository article Forcing over models of determinacy, written for the Handbook of Set Theory and available in DVI. and PDF.


Background reading for this workshop

Post-workshop materials:

Lecture notes from the workshop (PDF; Revised 10/09)

List of participants in this workshop